UNIVERSIDADE FEDERAL DE SANTA CATARINA CENTRO TECNOLÓGICO
AUTOMATION AND SYSTEMS ENGINEERING GRADUATE PROGRAM
Alice Ferreira Branco
Modelling and Control of a Thermal Solar Energy Generation Process
Florianópolis 2019
Alice Ferreira Branco
Modelling and Control of a Thermal Solar Energy Generation Process
Dissertation presented to the Automation and Systems Engineering Graduate Program in partial fulfillment of the requirements for the degree of Master in Automation and Sys-tems Engineering.
Advisor: Prof. Dr. Julio Elias Normey-Rico Co-advisor: Prof. Dr. Gustavo Artur An-drade
Florianópolis 2019
Ficha de identificação da obra elaborada pelo autor,
através do Programa de Geração Automática da Biblioteca Universitária da UFSC.
Ferreira Branco, Alice
Modelling and control of a thermal solar energy generation process / Alice Ferreira Branco ; orientador, Júlio Elias Normey-Rico, coorientador, Gustavo Artur Andrade, 2019.
107 p.
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2019. Inclui referências.
1. Engenharia de Automação e Sistemas. 2. Energia solar. 3. Geração renovável de energia. 4. Modelagem de sistema. 5. Controle preditivo. I. Elias Normey-Rico, Júlio. II. Artur Andrade, Gustavo. III. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia de Automação e Sistemas. IV. Título.
Alice Ferreira Branco
Modelling and Control of a Thermal Solar Energy Generation Process
The present work in the master level was evaluated and approved by the examining board composed by the following members:
Prof. Dr. Marcelo De Lellis Costa de Oliveira Universidade Federal de Santa Catarina
Dr. Rafaela Frota Reinaldo Petrobras
(Video conference)
Prof. Dr. Daniel Martins Lima Universidade Federal de Santa Catarina
This Dissertation is recommended in partial fulfillment of the requirements for the degree of “Master in Automation and Systems Engineering”, which has been approved in its present form by the Automation and Systems Engineering Graduate Program.
Prof. Dr. Werner Kraus Junior Graduate Program Coordinator
Prof. Dr. Julio Elias Normey-Rico Advisor
Florianópolis, August 30th 2019. Assinado de forma digital por Werner Kraus Junior:53108523953 Dados: 2019.12.05 09:45:51 -03'00'
Documento assinado digitalmente Julio Elias Normey Rico Data: 05/12/2019 10:26:52-0300 CPF: 762.840.859-15
This thesis is dedicated to former presidents Lula and Dilma, who invested in the Brazilian public universities.
Acknowledgements
First and foremost I want to thank my family. When I was younger, I wanted to become a doctor before I turned thirty years old to "beat" my father. Thanks to my parents, who have given me all the comfort and opportunity to study, I might be able to accomplish that dream. They were always my role models: my father as an amazing professor and my mother and sister as the greatest and strongest women I know. I hope this thesis makes you proud.
My sincerest thanks to my tutors Prof. Dr. Julio Elias Normey-Rico and Prof. Dr. Gustavo Artur Andrade, who were always available and willing to teach and guide me through this thesis. I aim to become as good of a professor as you.
Thank you to my colleagues, with whom I had many discussions about control and optimization theory, politics and trivial matters. You have made my working days easier and lighterxf.
A special thanks to my close friends and boyfriend, who suffered through my self doubt and mood swings. I can honestly say that I would not be able to complete this thesis without you.
This master thesis was supported financially by the Coordena¸cão de Aperfei¸coamento de Pessoal de Nível Superior (CAPES) and by the Centro de Pesquisas Leopoldo Américo Miguez de Mello (CENPES) which is the main research center from Petrobras. The work here described was developed in a project by Petrobras/ANEEL in conjunction with the Departamento de Automa¸cão e Sistemas (DAS). The name of the project is "Aprimoramento e valida¸cão de plataforma de simula¸cão de plantas heliotérmicas de concentra¸cão linear com estudos de inova¸cão tecnológicas" and the professor responsible for the control portion of the project is Prof. Dr. Julio EliasNormey-Rico.
In the beginning the Universe was created. This has made a lot of people very angry and has been widely regarded as a bad move.
RESUMO
Esta disserta¸cão de mestrado apresenta a modelagem de um processo de gera¸cão de energia solar térmica e o algoritmo de controle avan¸cado aplicado ao modelo para ma-ximizar a eficiência energética. É simples encontrar na literatura bons modelos de cada equipamento do processo de energia solar, porém não são encontrados muitos artigos que focam na modelagem de todo o processo, especialmente o processo utilizado neste trabalho, com um sistema de gera¸cão indireta de calor, que contém: um campo de coletores solares, dois tanques de armazenamento, um aquecedor a gás, um gerador de vapor, uma turbina a vapor, um gerador de potência elétrica e um condensador. Assim, uma modelagem completa de cada subsistema foi realizada e todos os modelos foram reunidos para simular o processo completo. É importante ter uma modelagem satisfató-ria do sistema, para que a mesma descreva a sua dinâmica, mas que não seja complexa demais para simula¸cão em um computador pessoal ou em um microcontrolador. O modelo do sistema é essencial, especialmente devido à técnica de controle usada para controlar o sistema: Modelo Preditivo Baseado em Modelo, MPC, que usa a predi¸cão da saída do sistema, baseada no seu modelo, para calcular a a¸cão de controle. MPC foi usado para controlar o processo com a inten¸cão de otimizar a energia gerada pelo sistema. Uma máquina de estados também foi implementada para a decisão de quais equipamentos devem estar ligado, dependendo da sua condi¸cão de temperatura, com o objetivo de evitar o consumo desnecessário de energia. Resultados de simula¸cão com dados reais medidos de uma simula¸cão de quinze horas são apresentados para retratar como a máquina de estados e as estratégias de controle escolhidas funcionam no modelo selecionado. Duas simula¸cões diferentes são feitas: primeiro, o processo é simulado com seus controladores locais, na segunda simula¸cão são adicionados MPC para otimizar a energia gerada, manipulando a pressão de referência do gerador de vapor, a abertura da válvula da turbina a vapor e a potência aplicada pelo aquecedor a gás. Os resultados da modelagem são satisfatórios, assim como o dos controladores, seguindo a referência desejada e gerado a energia necessária.
Palavras-chave: Energia solar. Gera¸cão renovável de energia. Modelagem de sistema. Controle preditivo.
RESUMO EXTENDIDO
Introdu¸cão
Para obter um desenvolvimento sustentável ideal, o foco para gera¸cão de energia nos próximos anos será em energias renováveis e limpas geradas de maneira eficiente. Sis-temas de gera¸cão de energia com combustíveis fósseis ainda são prevalentes, porém eles têm caráter limitado. Como há cada vez mais aumento da demanda de energia, o mundo está investindo em tecnologias de gera¸cão de energia baseadas em fontes renováveis, que serão uma parcela importante da produ¸cão global de energia nos pró-ximos anos, visando à redu¸cão do impacto ambiental, o aumento da oferta de energia e a diversifica¸cão da matriz energética. Para substituir combustíveis fósseis, energias renováveis devem competir em termos de eficiência, custo e viabilidade. Dessa maneira, pesquisas científicas e tecnologias estão crescendo nos últimos anos para aumentar a eficiência de sistemas de gera¸cão de energia renovável do ponto de vista de controle e otimiza¸cão.
O Brasil está pesquisando para melhorar a capacidade de gera¸cão renovável, principal-mente em energia solar, eólica e da biomassa. Dentre as diversas fontes renováveis, o Brasil tem um grande potencial para o aproveitamento de energia solar e a produ¸cão de energia elétrica a partir dessa fonte primária. Devido às características particulares do clima do Brasil, os processos e sistemas de controle a serem implementados devem ser adaptados para o melhor aproveitamento da energia. Neste sentido, técnicas de controle avan¸cado, modelagem e otimiza¸cão terão que ser desenvolvidas especificamente para esse tipo de processo, o que gera para a indústria brasileira um conhecimento específico para o desenvolvimento de tecnologia local.
Plantas solares térmicas possuem processos complexos multivariáveis, com grandes atrasos de transporte dos sistemas térmicos, várias restri¸cões de opera¸cão da planta e diversos subsistemas, em que cada estrutura tem dinâmicas, não linearidades e incerte-zas. É necessário ter uma boa modelagem do processo para aplica¸cão de controladores. O modelo deve ser complexo o bastante para representar bem suas características e ao mesmo tempo deve ser simples o suficiente para o cálculo da a¸cão de controle e para evitar excesso de esfor¸cos computacionais.
Uma op¸cão de controle avan¸cado para plantas industriais é o Controle Preditivo Baseado em Modelo, um tipo de controle que busca minimizar uma certa fun¸cão objetivo em um horiznte de predi¸cão. Para isso, é utilizado um modelo do processo que permita calcular as predi¸cões do comportamento futuro da planta. Assim, as a¸cões de controle futuras são calculadas usando técnicas de otimiza¸cão em tempo real, as quais permitem manter o sistema operando dentro de um conjunto de restri¸cões.
O MPC está sendo cada vez mais utilizado na indústria de energias renováveis, princi-palmente devido à possibilidade de adicionar restri¸cões ao controle. A modelagem de plantas solares térmicas e MPC aplicado às mesmas são o tema de estudo deste trabalho.
Objetivos
Objetivos gerais
O objetivo geral deste trabalho é estudar, simular e modelar os principais equipamentos de uma planta solar baseada na tecnologia cilindro-parabólico: camplo solar, tanques de armazenamento, gerador de vapor, aquecedor a gás, turbina e condensador. Além disso, será proposto um método de controle preditivo para otimizar a produ¸cão de energia através da planta solar térmica, considerando demanda variável e gasto de combustível fóssil para suprir a demanda energética quando a gera¸cão da planta solar for insuficiente. Objetivos específicos
Para alcan¸car o objetivo geral, é necessário modelar cada equipamento separadamente, e depois unir os modelos para simular o sistema completo: campos solares, tanques de armazenamento, gerador de vapor, aquecedor a gás, turbina a vapor, gerador de energia e condensador; projetar as máquinas de estados com modos de opera¸cão para acionar somente os equipamentos necessários, dependendo da condi¸cão de temperatura de cada um, para evitar a perda desnecessária de energia; e, por fim, implementar e simular es-tratégias de controle para otimizar a gera¸cão de energia considerando demanda variável e uso de combustível fóssil pelo aquecedor a gás.
Metodologia
O projeto consiste em uma revisão bibliográfica de técnicas de MPC aplicadas em plantas de energia solar térmica, modelagem de sistemas de gera¸cão de energia, implementa¸cão de MPC considerando as restri¸cões do processo e simula¸cão do processo com os controles calculados.
Cada equipamento da planta foi modelado baseado em um modelo encontrado na lite-ratura, com o intuito de achar o modelo mais simplificado para o processo, diminuindo esfor¸cos computacionais da simula¸cão e do cálculo de controle, mas ao mesmo tempo mantendo as características dinâmicas e estáticas fundamentais do processo.
Uma máquina de estados, com modos de opera¸cão, foi desenvolvida, com o propósito de definir qual equipamento estará ligado em determinadas condi¸cões, para evitar o gasto desnecessário de energia.
Para cada modo de opera¸cão, foram aplicadas diferentes estratégias de controle na simula¸cão. Foram estudadas formula¸cões de MPC para otimizar a produ¸cão de energia solar, considerando demanda variável e possível consumo de combustível fóssil pelo aquecedor a gás. Após o estudo, a estratégia de controle selecionada foi implementada e simulada com os modelos escolhidos.
Com dados reais de perturba¸cões, o processo foi simulado com os controladores imple-mentados e uma análise da resposta do sistema foi realizada.
Resultados e Discussão
Duas simula¸cões são realizadas para a análise do processo e controles aplicados. Pri-meiramente, uma simula¸cão do processo com os controladores locais é apresentada. Os controladores são capazes de manter cada equipamento no seu respectivo ponto de opera¸cão e, em grande parte da simula¸cão, conseguem gerar energia elétrica suficiente para os dados apresentados.
Após a simula¸cão com controladores locais, um controle avan¸cado é adicionado ao sistema para gerar a energia necessária. O controlador calcula a abertura da válvula da turbina a vapor e a pressão de referência do gerador de vapor para otimizar a gera¸cão de energia. Ao final da simula¸cão, quando a irradia¸cão come¸ca a diminuir, o aquecedor a gás é ligado para aumentar a temperatura do óleo na entrada do gerador de vapor. O aquecedor a gás utiliza combustível fóssil e seu uso deve ser minimizado. Por isso, ele só é ligado quando a abertura da válvula e a vazão de saída do tanque quente, usada para controlar a pressão de saída do gerador de vapor, estão nos seus respectivos valores máximos. Sua potência é calculada por outro controlador avan¸cado. Novamente, o resultado é satisfatório e a simula¸cão mostra que o controle aplicado no processo é capaz de gerar a energia necessária.
Considera¸cões Finais
O trabalho apresenta resultados satisfatórios, tanto para a modelagem quanto para os controladores avan¸cados. O processo não necessita de um aquecedor a gás, caso seja mantido na sua região de opera¸cão, porém, caso haja muitas nuvens bloqueando a irradia¸cão do sol ou a demanda de energia cres¸ca muito, o aquecedor torna-se necessário para aumentar a temperatura de saída do tanque quente e continuar gerando a energia necessária.
Palavras-chave: Energia solar. Gera¸cão renovável de energia. Modelagem de sistema. Controle preditivo.
ABSTRACT
This master’s dissertation presents a modelling of a solar thermal energy generation process and the advanced control algorithm applied to it to maximize energy efficiency. It is easy to find in the literature a good model for each subsystem of a solar energy process, but there aren’t many articles which focus on the modelling of the whole pro-cess, especially the one used in this work, with indirect solar heating system: a solar collector field, two storage tanks, a gas heater, a steam generator, a steam turbine, an electricity generator and a condenser. Thus, a complete modelling of each subsystem was conducted and brought together to simulate the entire process. It is important to have a satisfactory modelling of the system, so that it describes its dynamics, but which is not too complex to simulate on a personal computer or a microcontroller. The system’s model is essential especially because of the control technique used to control the system: Model Predictive Control, MPC, which uses the prediction of the system’s output based on its model to calculate the control law. MPC was used to control the process with the intent to optimize the energy generated by the system. Also, a state machine was im-plemented to decide which equipment will be turned off, depending on its temperature condition, aiming to avoid unnecessary energy consumption. Simulation results with real measured data are presented to depict how the state machine and control strategies chosen will work on the selected model. Two separate simulations are done: first, the process is simulated with its local controllers, to validate the chosen models; then an MPC is added to optimize the energy generated, by manipulating the steam generator’s output pressure reference and the steam turbine’s valve opening, with a second MPC do manipulate the gas heater applied power. The modelling results are satisfactory, much like the advanced controllers, tracking the desired reference and generating the neces-sary energy. For a determined operating range, it is not necesneces-sary to use the gas heater since it is fossil fuel, but if the irradiation decreases too much or the power demand is too high, it should be used to supply the power demand.
Keywords:Solar energy. Renewable energy generation. System modelling. Predictive Control.
LIST OF FIGURES
1.1 Brazil’s energy matrix distribution in 2018, adapted from (Secretaria de
Planejamento e Desenvolvimento Energético, 2019). . . 2
1.2 Brazil’s electrical energy matrix distribution in 2018, adapted from (Secre-taria de Planejamento e Desenvolvimento Energético, 2019) . . . 2
1.3 Direct Normal Irradiation in the Brazilian territory (??) . . . 3
2.1 Simplified configuration of a thermal solar energy generation process with indirect steam generation . . . 7
2.2 Entire process modelled for this project . . . 9
2.3 Schematic drawing of a parabolic trough solar collector (GHASEMI; RAN-JBAR, 2016) . . . 11
2.4 Schematic of the pipe depicting the discretization method (POWELL; EDGAR, 2012) . . . 13
2.5 Schematic of a storage tank divided in three sections (PASAMONTES et al., 2011) . . . 14
2.6 Schematic of a steam generator, adapted from (LI et al., 2019) . . . 16
2.7 Schematic of the steam turbine subsystems (KULKOWSKI; KOBYLARZ; DUZINKIEWICZ, 2015) . . . 19
2.8 Final connected process . . . 28
3.1 General strategy of an MPC (CAMACHO; BORDONS, 2007) . . . 31
3.2 Basic MPC structure (BRANDãO, 2018) . . . 31
3.3 Free and forced response (CAMACHO; BORDONS, 2007) . . . 33
4.1 State machine . . . 45
4.2 Solar collector field operating in mode 1 . . . 46
4.3 Solar thermal plant operating in mode 2 . . . 47
4.4 Process operating in mode 3 . . . 48
4.5 Process operating in mode 4 . . . 49
4.6 Storage tanks and electrical power generation section operating in mode 5 . 50 4.7 Block diagram of the control strategy applied in mode 1 . . . 51
4.8 Block diagram of the control strategy applied in mode 2 . . . 53
4.9 Block diagram of the advanced control strategy loop of the process . . . 55
4.10 Block diagram of the advanced control strategy loop of the process with gas heater . . . 58
5.1 Values of the main disturbances through time for simulation 1 . . . 62
5.2 Values of the main disturbances through time for simulation 2 . . . 63
5.3 Heat generation section’s behaviour and operating modes with local con-trollers in simulation 1 . . . 64
5.4 Steam generator’s output pressure behaviour with local controllers in simu-lation 1 . . . 66
5.5 Steam generator’s drum level behaviour with local controllers in simulation 1 67
5.6 Steam turbine and condenser’s behaviour with local controllers in simulation 1 68
5.7 Energy generating section’s behaviour with local controllers in simulation 1 69
5.8 Gas heater’s behaviour with local controllers in simulation 1 . . . 70
5.9 Heat generation section’s behaviour and operating modes with local con-trollers in simulation 2 . . . 71
5.10 Steam generator’s behaviour and operating modes with local controllers in simulation 2 . . . 72
5.11 Steam generator’s drum level behaviour with local controllers in simulation 2 73
5.12 Energy generating section’s behaviour with local controllers in simulation 2 74
5.13 Condenser’s behaviour with local controllers in simulation 2 . . . 75
5.14 Heat generating section behaviour and operating modes with advanced con-trollers in simulation 1 . . . 77
5.15 Steam generator’s output pressure behaviour with advanced controllers in simulation 1 . . . 78
5.16 Gas heater’s behaviour with advanced controllers in simulation 1 . . . 79
5.17 Steam generator’s drum level behaviour with advanced controllers in simu-lation 1 . . . 80
5.18 Steam turbine and condenser’s behaviour with advanced controllers in simu-lation 1 . . . 81
5.19 Energy generating section’s behaviour with advanced controllers in simula-tion 1 . . . 82
5.20 Steam generator’s output pressure behaviour with advanced controllers in simulation 2 . . . 83
5.21 Steam generator’s water level behaviour with advanced controllers in simu-lation 2 . . . 84
5.22 Electrical power generation section’s behaviour with advanced controllers in simulation 2 . . . 85
5.23 Condenser’s behaviour with advanced controllers in simulation 2 . . . 86
LIST OF TABLES
2.1 Variables of each equipment . . . 29
4.1 Table of operating modes . . . 43
A.1 Table of solar collectors field parameters . . . 99
A.2 Table of storage tanks parameters . . . 99
A.3 Table of gas heater parameters . . . 100
A.4 Table of control valve parameters . . . 100
A.5 Table of high pressure stages dead space parameters . . . 100
A.6 Table of high pressure stages parameters . . . 101
A.7 Table of high pressure stages parameters . . . 102
A.8 Table of moisture separator parameters . . . 102
A.9 Table of reheater parameters . . . 103
A.10 Table of low pressure stages dead space parameters . . . 103
A.11 Table of low pressure stages parameters . . . 104
A.12 Table of power parameters . . . 105
A.13 Table of power parameters . . . 106
ACRONYMS
DSG Direct Steam Generator
HP High Pressure
HTF Heat Transfer Fluid ISG Indirect Steam Generator
LMPC Linear Model Predictive Control
LP Low Pressure
MPC Model Predictive Control
NMPC Non-linear Model Predictive Control ODE Ordinary Differential Equation
P Proportional
PDE Partial Differential Equation PI Proportional Integral
PNMPC Practical Non-linear Model Predictive Control
SYMBOLS A Area CV Control valve C Thermal capacity DS Dead space HP High pressure I Integral LP Low pressure M S Moisture separator N Power P Pressure R Reheater T Temperature V Volume α Valve opening ˙
m Massic flow rate
˙q Volumetric flow rate
η Parameter ρ Density a Ambient c Condenser d Diameter e External gh Gas heater ht Hot tank h Enthalpy in Input i Internal k Gain l Level m Metal nom Nominal n Horizon out Output o Oil
ref Reference
scf Solar collector field
sg Steam generator st Steam turbine s Steam t Time u Control v Velocity w Water
CONTENTS 1 Introduction 1 1.1 Objectives . . . 4 1.1.1 General objective . . . 4 1.1.2 Specific objectives. . . 4 1.2 Problem statement . . . 5
1.3 Organisation of the dissertation . . . 5
2 System Modelling 7 2.1 System Modelling . . . 7
2.1.1 Solar Collectors Field . . . 11
2.1.2 Storage Tank . . . 13
2.1.3 Gas Heater . . . 14
2.1.4 Steam Generator . . . 15
2.1.5 Steam Turbine. . . 18
2.1.5.1 Control valve . . . 19
2.1.5.2 Dead space of the High Pressure Stages . . . 20
2.1.5.3 High Pressure Stages . . . 21
2.1.5.4 Moisture Separator . . . 22
2.1.5.5 Reheater . . . 22
2.1.5.6 Dead space of the Low Pressure Stages . . . 23
2.1.5.7 Low Pressure Stages . . . 24
2.1.5.8 Mechanical Power . . . 24
2.1.6 Electrical Power Generator . . . 25
2.1.7 Condenser . . . 26
2.2 The final process . . . 27
2.3 Final Remarks . . . 29
3 Model Predictive Control 30 3.1 Linear Model Predictive Control . . . 33
3.1.1 Free and forced response . . . 33
3.1.2 Prediction error correction . . . 34
3.2 Objective function . . . 34
3.3 Control law . . . 36
3.4 Non-linear Model Predictive Control . . . 36
3.4.1.1 Decomposition Technique for Calculation of the
Non-linear Control Law . . . 38
3.4.2 Practical Non-linear Model Predictive Control . . . 38
3.4.2.1 PNMPC’s Algorithm . . . 39
3.4.2.2 PNMPC’s prediction error correction . . . 40
3.5 Final Remarks . . . 41
4 Operating modes and control strategies 42 4.1 Operating modes . . . 42 4.1.1 Mode 0 . . . 45 4.1.2 Mode 1 . . . 46 4.1.3 Mode 2 . . . 46 4.1.4 Mode 3 . . . 47 4.1.5 Mode 4 . . . 48 4.1.6 Mode 5 . . . 49
4.2 Control strategy applied to each operating mode. . . 51
4.2.1 Mode 0 . . . 51
4.2.2 Mode 1 . . . 51
4.2.3 Mode 2 . . . 52
4.2.4 Mode 3 . . . 53
4.2.4.1 PNMPC applied to the solar plant . . . 54
4.2.5 Mode 4 . . . 57
4.2.5.1 PNMPC applied to the gas heater . . . 58
4.2.6 Mode 5 . . . 59
4.3 Final Remarks . . . 60
5 Results and Discussion 61 5.1 Disturbance data . . . 61
5.2 Results with local controllers and operating modes . . . 63
5.2.1 Simulation 1. . . 63
5.2.2 Simulation 2. . . 69
5.3 Results with advanced controllers and operating modes strategy . . . 74
5.3.1 Simulation 1. . . 75
5.3.2 Simulation 2. . . 83
5.4 Comparison . . . 86
5.4.2 Simulation 2. . . 88 5.5 Final remarks . . . 89 6 Conclusions 90 6.1 Contributions . . . 91 6.2 Future Works . . . 91 References 92 A Parameters Tables 99
A.1 Solar Collectors Field . . . 99
A.2 Storage Tanks . . . 99
A.3 Gas Heater. . . 100
A.4 Steam Turbine. . . 100
A.4.1 Control Valve . . . 100
A.4.2 High Pressure Dead Space . . . 100
A.4.3 High Pressure Stages . . . 101
A.4.4 Moisture Separator . . . 102
A.4.5 Reheater . . . 103
A.4.6 Low Pressure Dead Space . . . 103
A.4.7 Low Pressure Stages . . . 104
A.4.8 Power . . . 105
1 INTRODUCTION
To attain sustainable development, the focus for energy generation on the forth-coming years will be on renewable and clean energy generation, which has to be achieved efficiently. Although fossil fuelled energy generation systems are still used broadly, the use of renewable sources has been growing each year aiming to reduce the environmental impact of energy generation (CAMACHO; BERENGUEL,2012). In this context, renewable energy generation systems must compete in terms of efficiency, cost and viability, in order to replace fossil fuels.
The demand for energy is growing each day (SHAFIEE; TOPAL,2009), however fuel prices are rising, due to future scarcity (SHAFIEE; TOPAL, 2010). This is leading many countries in a search for alternative energy sources instead of the common, fossil fuelled ones (JOHANSSON et al., 1993). One of these alternative energy sources is renewable energies, which allow sustainable and ecological development, though there are still too expensive and can not compete directly with fossil fuel generation. Yet, with further studies, renewable energy may be a cheaper alternative for energy generation (MORATO et al.,2018).
There is a global effort on the study of renewable energies (CAMACHO; BERENGUEL,
2012), so that they will become viable energy sources on a large scale generation sce-nario. The growing interest on the study of renewable energies may be exemplified: in (MO; SANSAVINI, 2018), a real time coordination of energy generation based on renewable sources with frequency control was studied. In (PARISIO; VECCHIO; VAC-CARO,2012), an optimisation problem, focusing on robustness for the management of renewable energy systems is presented. In (ALARCON-RODRIGUEZ; AULT; GALLOWAY,
2010), a state of the art general panorama was studied of a multi-objective planning of distributed solar thermal plants. Thus, scientific and technological researches are growing to increase the efficiency of renewable energy generation systems from the point of view of control engineering and optimisation.
In 2015, only 7.1% of the total energy generated in the world came from renew-able sources (AGENCY, 2017). Although that is a small value, it still represents an increase of ten times comparing to 1973. Yet, in Brazil, this value can reach up to 45%, deriving from several sources: sugar cane bagasse, solar, hydro-electric, as can be seen in Figure1.1, where the values are based on the year of 2018.
Chapter 1. Introduction 3
Figure 1.3: Direct Normal Irradiation in the Brazilian territory (??)
Thermal solar plants are complex processes with several subsystems, each one having different dynamics, non-linearities and distinct uncertainties. To optimize the energy generation process, there’s a need to use advanced control strategies, such as
Model Predictive Control (MPC) (MACIEJOWSKI,2002). To do so, it is fundamental to obtain an accurate, realistic model of the system.
The plant modelled in this work contains several subsystems for the electrical energy generation: solar collectors, storage tanks, steam generator, gas heater, steam turbine, energy generator and condenser. While many different models of each subsys-tem are found in the literature, there are not many studies which combine all the models to simulate the whole process using solar irradiation as the main energy source with the focus of control studies, thus, focusing on modelling the system’s dynamics in a simple but efficient manner with its manipulated, process and disturbances variables.
A good model of the system is necessary for a better control calculation. The model must be complex enough to be a good representation of the process characteristics, being non-linearities, non-minimal phases or multi variable systems, while at the same time being simple enough for the control action calculation and to avoid excessive
Chapter 1. Introduction 4 computational efforts.
A viable option to control solar thermal plants is theMPCstrategy, which is a type of advanced control with several different formulations which differ amongst themselves in the model used to represent the process, the disturbances and the cost function. The common steps of theMPCalgorithms can be seen in (BORRELLI; BEMPORAD; MORARI,
2017).
While currentlyMPCstrategies have a consolidated theoretical base in the academia, they had their origin in the industry with little theoretical knowledge behind the method-ology used (PATWARDHAN,2014). MPCis a group of advanced control techniques used in the industry, including the renewable energy industry, mainly because they can be applied in complex processes, such as multi variable, non-minimal phase, non-linear processes. MPCis also largely used since it considers the restrictions of the system and is useful when future references are known (CAMACHO; ALBA,(2013)).
Thus, having in mind the current context of transition between fossil fuels and clean energies and given what has been presented about control of renewable processes, this work proposes an adequate model of a thermal solar plant which generates electric energy, by combining known and tested models of each system from the literature. Moreover, a complete control strategy based on anMPCand state machine is proposed to optimize the plant operation. Simulation results are used in this work to validate the model and controller and to compare the obtained results with other solutions, such as plants operation without advanced control.
1.1 OBJECTIVES
1.1.1 General objective
The general objective of this master’s project is to study, simulate and model each subsystem of a solar plant. Besides that, it will also be considered the implementation of a predictive control method to optimize the energy production through a thermal solar plant, considering electrical energy variable demand.
1.1.2 Specific objectives
To reach the general objective, the following specific objectives are listed:
• Addition of the model of the steam generator, steam turbine, energy generator, condenser and gas heater to an existing simulation of the solar collectors field and the storage tanks;
Chapter 1. Introduction 5 • Design of a state machine which changes the plant’s operating modes to turn on
only the necessary equipment to avoid unnecessary use of energy;
• Implemention and simulation of control strategies to optimize the energy genera-tion considering variable demand and fossil fuel usage of the gas heater.
1.2 PROBLEM STATEMENT
The project consists of a literature review ofMPCapplied to solar energy plants, modelling of energy generation systems, MPCimplementation considering the process restrictions and simulation of the process with the calculated controllers:
• System modelling: Models of each system of the plant were investigated in the literature, aiming for a linear and simple modelling of the process, to diminish computational efforts on the simulation and the control calculation;
• Control strategies: Operation modes were developed, with the purpose of defin-ing which subsystems will be turned on in specific conditions, to avoid unnecessary energy spending. For each operation mode, different control strategies were ap-plied in the simulation. It was studied in the literature MPC formulations to optimize the production of energy, considering a variable demand and possible fossil fuel consumption by the gas heater. Then, the chosen control strategies were implemented and simulated;
• System simulation: With real disturbance data, the process was simulated with the implemented controllers and an analysis of the system’s response was done.
1.3 ORGANISATION OF THE DISSERTATION This dissertation is organised as follows:
Chapter2 describes the energy generation process and the modelling done of each subsystem of the thermal solar process. The process modelled in this thesis is of indirect steam generation, which uses a heat transfer fluid, HTF, to absorb heat from the sun’s irradiation and to transfer it to the water to vaporize it and generate steam. Chapter2 is divided in two sections: the heat generation section, where the oil flows, and the energy generation section, where the water and steam flow. The model for each equipment was chosen from literature and then brought together to simulate the whole process and calculate the optimum control strategy to optimize the energy generated.
Chapter 1. Introduction 6 Chapter 3 presents the theoretical background of MPC strategies, focusing on non-linear MPC. MPC is used in this thesis since it is an advanced control strategy proven to function well in industrial process and because it calculates the control action considering the system’s restrictions. The main steps for allMPCare presented, where a predictive model is calculated to optimize an objective function subject to constraints. A Non-linearMPCstrategy, thePractical Non-linear Model Predictive Control(PNMPC) is presented, which is applied to the process to optimize the energy generation based on the model presented in Chapter2.
The control strategy proposed to solve the solar energy generation problem is introduced in chapter4, including the operation modes, local and advanced controllers. For the system to work properly, local controllers are added to the equipment to maintain them at their respective operation point. To avoid wasted energy, operation modes are calculated and added to the model. On each operation mode, some equipment are turned on or off, depending on irradiation and outlet temperature of the equipments. TwoPNMPCare calculated and applied to optimize the energy generated by the system. The results and discussion of the implemented control algorithm are discussed in chapter 5. Using data for a fifteen hour simulated operation, the process is simulated when controlled with the local controllers and the operating mode strategy. Finally, a simulation of the process with the appliedPNMPCto optimize the energy generated is shown, both with and without gas heater.
The work ends at chapter6, where the conclusions, contributions of the work and suggestions of future works are shown.
Chapter 2. System Modelling 8 In solar thermal plants, solar collectors are needed to capture thermal energy from solar irradiation and anHTFis needed to transport the heat. It is also possible to integrate heat storage elements in the heat generation section, such as tanks, allowing the continuous operation of the process when the solar irradiation is not enough to generate the thermal energy for the system.
The power block is based on the Rankine cycle, which is a thermodynamic cycle of a heat engine that converts heat into mechanical power (KATSANOS; HOUNTALAS; PARIOTIS,2012). The fluid, which is generally water and the fluid used in this project, flows in a closed loop and is reused constantly by phase changes, which can be seen by the cycle explained below, divided in four main parts:
1. Circulation pump: utilizes power to pump the water from low to high pressure from the condenser to the steam generator. The pump requires very little input energy;
2. Steam generator: the steam generator, which is the high temperature heat source, receives the water in liquid phase with high pressure where it is heated by exchang-ing heat with the heated oil from the heat generation section to evaporate and become a dry saturated vapour;
3. Steam turbine: the power generator expands the dry vapour in the turbine to drive the electricity generator and generate electricity. The turbine decreases the vapour’s temperature and pressure. Some condensation may occur, thus there might be a moisture separator and reheater in this section;
4. Condenser: it is used as a low temperature heat sink, where the vapour enters and is condensed at a constant pressure to go back to liquid phase and be pumped again by the circulation pump.
The process for this work has eight separate equipment: the solar collector field, two storage tanks, a steam generator, the gas heater, the steam turbine, the electricity generator and the condenser. They are interconnected and work as can be seen in the diagram of Figure2.2.
The two sections of the process are represented in Figure 2.2, where the heat exchange process, which is composed by the solar collectors field, the hot and cold tanks, the gas heater and the steam generator is represented by the colours red and blue and the energy generation section is represented by the colours blue and green and contains the steam generator, the steam turbine, the electricity generator and the condenser.
Chapter 2. System Modelling 10 In the energy generation section, the water, which comes from the condenser, exchanges heat with the hot oil inside the steam generator, to turn into steam. The generated steam flows into the turbine, which drives the generator to generate electrical energy. After it has passed through the turbine, the steam flows through the condenser, so it will turn into liquid water again and to be pumped to re-enter the steam generator. A first principles modelling of each subsystem is done in this thesis to simulate the process and design advanced control strategies for it. Most of the models are found by using mass, energy or momentum balance on each subsystem of the process, and the inlet of the next subsystem is the output of the previous subsystem.
While there are many models in the literature of each subsystem, there are not many modelling done for the entire process and the models found in the literature of the entire process do not use the same equipments as in this work. In (BARIGOZZI et al., 2012) the process modelled includes a heliostat field, a solar tower receiver and a gas turbine, where the simulation codes to the heliostat field and tower receiver are done on a different software than of the gas turbine, which generates electricity. In (BÜRGER et al., 2019) a thermal energy supply for a building is modelled, which is composed of a concrete slab under the building to receive heat, two water temperature storages and a recooling tower. In (JAMEL; RAHMAN; SHAMSUDDIN, 2013) it is shown the integration of conventional power plants, such as steam, with solar plants and of unconventional power plants, such as geothermal. In (ROUX; BELLO-OCHENDE; MEYER, 2013) a process with a solar field, receiver, turbine, load, recuperator and compressor is presented, focusing on the Brayton cycle. In (CAMERETTI et al., 2015) a study is done to optimize the generation of electricity by a micro gas turbine using a hybrid solar plant, combined with the use of biogas. In (HERVÁS,2008) a solar thermal plant with a solar collector field, a storage tank and a steam generator are modelled much like it will be in this work, but it does not show the energy generation model. In (GARCíA; ÁLVAREZ; BLANCO, 2011) a model much like the one in this work is presented: it has a solar field and two storage tanks, where molten salt circulates, a steam generator, steam turbine, generator and cooling system, but the focus is not to have a model to apply controllers. Given the processes studied in the literature, this thesis aims to gather models from the literature for each equipment, separately, and interconnect them to simulate the entire process.
Chapter 2. System Modelling 11 2.1.1 Solar Collectors Field
The solar collectors field is the main source of energy to the clean energy gen-erator process. It is used to raise the fluid’s temperature, in this case oil, to supply the system’s heat demand. A small schematic of a solar collector panel can be seen in Figure
2.3.
Figure 2.3: Schematic drawing of a parabolic trough solar collector (GHASEMI; RAN-JBAR,2016)
The mirrors on the parabolic concentrator focus the irradiation of the sun on the receiver, which absorbs the heat and transfers the heat to the oil which is flowing inside the pipes in the collector (POWELL; EDGAR, 2012). The metal that constitutes the pipe goes through a treatment at its surface so it has high absorbing and low emission properties, to absorb high quantities of radiation while minimizing radiating heat loss (KALOGIROU,2013;MESSEL; BUTLER,1975).
For the modelling of the solar collectors field, there are several works done in the literature. In (SILVA et al., 2014) a similar model of the one used in this thesis is introduced, but with the addition of the angle of the solar collectors and its shaded areas. In (ODEH; MORRISON; BEHNIA,1998) a direct steam solar collector is modelled, with emphasis on the two-phase flow inside the collector, which does not occur in this work since oil is used and it does not change its phase. In (MARC et al., 2011) it is
Chapter 2. System Modelling 12 presented the modelling of a double-glazed solar collector, which leads to a 6th order Partial Differential Equation(PDE) model. The selected model for this thesis was chosen based on its simplicity over the ones presented above.
The modelling of the process is done by calculating the energy balances of the fluid and the metal. It is obtained assuming that the dominant course of heat transfer is metal-to-fluid conduction. Thus, the collector is modelled by two coupled equations, based on the model presented in (CAMACHO; BERENGUEL, 2012; ÁLVAREZ; YEBRA; BERENGUEL,2007).
Some assumptions are made in order to simplify the model (PASAMONTES et al.,
2013): the fluid is incompressible, its heat capacity and specific heat are constant; the effects of the pressure gradient fluctuations in the temperature of the thermal oil are neglected; and the pipe’s wall heat resistance is dropped out from consideration.
Equation (2.1) models the metal’s temperature dynamic and Equation (2.2) mod-els the fluid’s temperature.
ρmCmAe ∆Tm(t) ∆t =deπηI(t) −deπhe(Tm(t) −Ta(t)) −diπhi(Tm(t) −Tscf(t)), (2.1) ρoCoAi ∂Tscf(t, x) ∂t = −˙qscf(t)ρoCo ∂Tscf(t, x) ∂x +diπhi(Tm(t) −Tscf(t)), (2.2) The output oil temperature,Tscf(t), is calculated by the heat transfer between the tube’s metal and the oil and between the oil in the previous and the current section of the collector, thus depending on the oil’s flow rate, ˙qscf(t). The metal’s temperature,Tm(t), is also calculated by heat transfer, but between the metal and the ambient temperature,
Ta(t), between the oil and the metal and from the irradiation, I(t). The parameter values are adapted from (PASAMONTES et al.,2013) to fit this process’ dimensions and can be seen in Table A.1on AppendixA.
To account for the delay between the output temperature of the oil and its input flow rate, a third equation is added to the model. The momentum balance is added (ANDERSON,1995), which calculates the difference between the input and output flow rate, shown in Equation (2.3). For each section of the tube, ∂x, the difference between the flow rate on the previous instant and the flow rate at this instant is calculated to find the variation of the flow rate on time.
d˙qscf(t) dt = fAi − 2˙qscf(t) Aiρo ∂˙qscf(t, x) ∂x , (2.3)
Chapter 2. System Modelling 15 turned on to heat the oil and supply the needed energy. But it is an expensive and non-renewable source of energy, so its use should be avoided.
The gas heater model is a mixture of terms which can be adjusted accordingly to known data or through parameter tuning (PASAMONTES et al.,2013). Some terms are used to approximate the mean behaviour of the unknown parameter’s set related to the gas. This happens because of the lack of direct measurement of important data, since the only available measurements are the oil flow and the input and output temper-atures. There is no data of the architecture in the inside of the equipment, thus a mostly empirical model is delivered. The resulting model can be seen in Equation (2.6):
CoρoVgh
dTgh(t)
dt = UghAgh(Ta(t) −Tgh(t)) + Ugtof(R(t)Tgas−Tgh(t)) +Coρf˙qht(Tht(t) −Tgh(t)), (2.6)
Similarly to the solar collectors and storage tanks models, the gas heater’s oil output temperature,Tgh(t), is calculated by heat transfer. It calculates the heat transfer between the oil and the ambient temperature and by the the input oil temperature, which is the hot tank’s output temperature,Tht(t), which depends on the oil’s flow rate, ˙qht(t). To increase the oil’s temperature, the heat transfer between the oil’s temperature and the power applied by the gas heater, R(t), is added to the calculation. The parameters values are taken from (PASAMONTES et al., 2013) and can be seen at Table A.3 on AppendixA. The use of the gas heater will be minimised by an optimizer, explained in Chapter4.
2.1.4 Steam Generator
The steam generator process is complex, especially since it deals with a two-phase process and there is presence of steam below liquid level, which causes a shrink-and-swell phenomenon (MINHAJULLAH,2011). Even though the complexity of the system is considerable, its behaviour can be captured by global mass and energy balances for control purposes (ÅSTRÖM; BELL,1989).
In (HAMMER; MORIN,2014), a model with four equations of a two-phase pipe flow is shown but since its solving process is too complex for this work, it is not used. In (FRAIDENRAICH et al.,2013), a direct steam generator power plant is modelled, so although a steam generator is modelled, it is for a DSGplant. In (PROCK,1988), a 6th
order non-linear model of a steam generator is presented, and the one used in this work is a 4th order model.
Chapter 2. System Modelling 17 be calculated due to the efficiency of heat and mass transfer considered. It captures well the dynamics of the system due to power changes.
e21
dVsg,w(t)
dt + e22
dPsg(t)
dt = Qsg(t) +m˙w(t)hw−m˙ s(t)hs, (2.7) To model the dynamics due to water or steam flow rate changes, Equation (2.8) is added by calculating the global mass balance.
e11
dVsg,w(t)
dt + e12
dPsg(t)
dt =m˙w(t) −m˙ s(t), (2.8) It is necessary to control the steam generator’s water level, thus it dynamics’ need to be modelled. To do so, it is necessary to model the shrink and swell effect in the steam generator’s drum. Thus, it is considered the distribution and the mass and energy transfer between the steam and water. The mass and energy balance are calculated and brought together in Equations (2.9) and (2.10).
e32 dPsg(t) dt + e33 αsg(t) dt = Qsg(t) −αsg(t)hcondm˙cond, (2.9) e42 dPsg(t) dt + e34 αsg(t) dt + e44 Vsg,s(t) dt = ρs ts (V0sg,s −Vsg,s(t)) + hw −hf hcond qw(t), (2.10) To calculate the water level, it is necessary to calculate the water volume inside the steam generator,Vwd(t), is calculated using Equation (2.11).
Vwd(t) =Vwt(t) −Vdc− (1 −αv)Vr. (2.11)
The water level inside the steam generator, l(t), is calculated using Equation (2.12), which is the sum of the water volume inside the steam generator and the dis-placement due to changes of the steam-water ratio, divided by the area of the drum.
l(t) = Vwd(t) +Vsd(t)
Ad
(2.12) wherehf,hs,hcond andhw are the specific enthalpy of the water inside the steam genera-tor, steam, condensation and input water, respectively, andm˙condis the forced circulation
boiler flow rate. The coefficients eji depend on the values of the states and their
Chapter 2. System Modelling 18 2.1.5 Steam Turbine
The steam turbine converts energy from high pressure and high temperature steam to mechanical power. The steam enters the turbine and it accelerates the moving blades attached to the rotor. The kinetic energy from the acceleration is converted to shaft torque, which loads the electricity generator to generate electrical energy ( SOKÓL-SKI; RUTKOWSOKÓL-SKI; DUZINKIEWICZ,2016). The steam turbine model depends vastly on its size and how much power it should generate.
In (DULAU; BICA,2014), a dynamic model of a steam turbine is presented, which is a combination of several 1st order transfer functions. In (REPORT,1973), a variety of different steam turbines are modelled, but the only output is the generated power and in this work more is needed, like its temperature, flow rate and pressure output. In (SOKÓLSKI; RUTKOWSKI; DUZINKIEWICZ,2016), a simplified fuzzy model of a steam turbine is presented. The chosen model for this thesis was made based on the available input variables and the calculated variables from the model.
For the model presented in this work, the manipulated variables of the turbine model are the input pressure and valve opening and the output variables calculated through dynamic models are pressure, temperature, steam mass flow rate and the tur-bine’s generated power. The dynamic model of the turbine is divided in several sub-systems: dead space, used as accumulation spaces; stage groups on theHigh Pressure
(HP) andLPsections, where the blades are; moisture separator, used to improve steam properties; and reheater, used to increase the steam’s temperature without altering its pressure (KULKOWSKI et al.,2017). The general scheme of the turbine can be seen in Figure2.7.
Chapter 2. System Modelling 20
ODEis used, which can be seen on Equation (2.13).
τCV
dα0(t)
dt +α(t) =kCVα
i(t). (2.13)
The output pressure of the control valve,Pout
CV(t), and its mass flow rate, m˙outCV(t), output are calculated based on static models shown on Equations (2.14) and (2.15), respectively. They are respectively calculated as a proportion to their nominal values, to the opening of the valve and its nominal value.
Pout CV(t) =P nom,out CV α0(t) αnom,0, (2.14) ˙ moutCV(t) =m˙ nom,out CV P (t) Pnom α0(t) αnom,0, (2.15)
whereαi(t) is the reference valve opening degree. The parameters can be seen at Table A.4on AppendixA.
2.1.5.2 Dead space of the High Pressure Stages
The dead space before the HP andLP sections are accumulation spaces of the steam, characterized by temperature and pressure drops. For the inlet of theHPsection, the Equations seen in (2.16) are used to calculate the output pressure dynamics of the dead space, Pout
DS,HP(t), and the parameters are calculated from the data given by the turbine manufacturer (KULKOWSKI; KOBYLARZ; DUZINKIEWICZ,2015).
τDS,HP = VDS,HP HDS,HPm˙nomDS,HP,inϑDS,HP , τDS,HP dPout DS,HP(t) dt =kDS,HP(m˙ out DS,HP(t) −m˙ in DS,HP(t)). (2.16)
It is considered that there is no leak on the dead spaces, thus all steam that enters the dead space goes out to the HP or the LP section. To calculate the output mass flow rate and the temperature of the dead space,m˙ out
DS,HP(t) and TDSout,HP(t), static relationships are used, seen in Equations (2.17) and (2.18), respectively.
˙ minDS,HP(t) =m˙ out CV(t), ˙ moutDS,HP(t) =m˙ in DS,HP(t), (2.17) Tout DS,HP(t) =ADS(P (t)outDS,HP) BDS(P (t)outDS,HP)CDS, (2.18) where Min
DS,HP(t) is the mass flow rate on the inlet of the dead space. The parameters can be seen at Table A.5on AppendixA.
Chapter 2. System Modelling 21 2.1.5.3 High Pressure Stages
A more complete modelling of the system would involve a dynamic modelling of theHPsection. But to diminish the computational effort, a static model of theHPstages is applied, as its dynamic time is considerably low compared to the other components of the steam turbine. This section is considered to have ten stages, represented by j in the following equations, and two vents after stages three and six.
For the calculation of the output pressure for each stage,Pout
HP,j, Equation (2.19) is used. Pin HP,j(t) =P out DS(t), j = 1, Pin HP,j(t) =P out HP,j - 1(t), PoutHP,j(t) =P nom,out HP,j Pin HP,j(t) PnomHP,j,in . (2.19)
For the calculation of the output temperature for each stage, Tout
HP,j(t), Equation
(2.20) is used.
THP,jout (t) =AT(PoutHP,j(t))
BT(PoutHP,j(t))CT (2.20)
For the calculation of the output mass flow rate for each stage,m˙out
HP,j(t), Equation
(2.21) is used. The output mass flow rate is calculated differently for stages with and without vents. ˙ moutHP,j(t) =m˙ nom,out HP,j \sqrt{} TnomHP,j,in Tin HP,j(t) Pin HP,j(t)2−PoutHP,j(t)2
(PnomHP,j,in)2− (Pnom,out HP,j )2
(2.21) For each stage with a vent, the calculation of the output mass flow rate for each vent, m˙out
HP v,j(t), Equation (2.22) is used, in which it is assumed a proportional steam
distribution into the vent and the next stage.
˙ moutHP v,j(t) =m˙ nom,out HP v,j ˙ mout HP,j=jvent(t) ˙
mnomHP,j,out=jvent , (2.22)
wherePin
HP,j(t),PoutHP,j - 1(t) are the pressure on the input of each stage and on the output
of the stage before, respectively, Tin
HP,j(t) is the input temperature on each stage and ˙
mout
HP,j=jvent(t) is the output mass flow rate of the stage equivalent to the vents’ stage. The
Chapter 2. System Modelling 22 2.1.5.4 Moisture Separator
The moisture separator separates liquid water from the steam generated, so the pressure, temperature and mass flow rate variables are calculated for the steam, rep-resented by the superscript S on the following equations, and for the liquid water, represented by the superscript W on the following equations.
The output mass flow rate of the moisture separator, m˙ S,outM S (t) is calculated by Equations (2.23). ˙ mS,inM S(t) =m˙ out HP,j(t), ˙ mS,outM S (t) =m˙S,inM S(t)x in xout, (2.23)
where m˙S,inM S(t) and m˙W,inM S (t) are the mass flow rate of the steam on the inlet of the moisture separator and of the water on the inlet of the moisture separator and TW,in
M S (t) is the water temperature on the inlet of the moisture separator. The parameters can be seen at TableA.8on Appendix A.
The water output temperature of the moisture separator, TW,out
M S (t) is calculated by Equation (2.24).
TM SW,in(t) = Tout HP,j(t),
TM SW,out(t) = TM Snom,W,out(PT3
TM SW,in(t) TM Snom,W,in
+PT4)
(2.24)
The output pressure of the moisture separator, PS,outM S (t) andP W,out
M S (t) are equal to the output pressure at the final stage of the high pressure section since the moisture separator does not cause any pressure or temperature drop on the steam.
2.1.5.5 Reheater
To increase the power generated by the turbine, it is necessary to raise the steam temperature, which is done by the reheater. The calculation of the output mass flow rate of the reheater,m˙out
R (t), is done accordingly to the Equations (2.25).
˙ min R(t) =m˙ S,out M S (t), ˙ moutR (t) =m˙ nom,out M S ˙ min R(t) ˙ mnomR ,in (2.25)
Chapter 2. System Modelling 23 The calculation of the output pressure of the reheater,Pout
R (t), is done according to Equation (2.26). Pin R(t) =P S,out M S (t), τR= VR
HRavgMRnom,outυR Pout R (t) PnomR ,out 1−HR HR , τR dPout R (t) dt =kR(m˙ in R(t) −m˙ out LP,1(t)) (2.26)
The calculation of the output temperature of the reheater, Tout
R (t), is done accord-ing to Equations (2.27). TRout(t) = T nom,out R (HT + IT Pin R(t) PnomR ,in), (2.27) wherem˙in
R(t) andm˙ outLP,1(t) are the mass flow rate at the input of the reheater and at the
output of the first group of stages on theLPsection, respectively. Pin
R(t) is the pressures
on the input of the reheater. The parameters can be seen at Table A.9on AppendixA.
2.1.5.6 Dead space of the Low Pressure Stages
For the inlet of the LP section, Equations seen in (2.28) are used to calculate the output pressure dynamics of the dead space, Pout
DS,LP(t), and the parameters are calculated from the data given by the turbine manufacturer (KULKOWSKI; KOBYLARZ; DUZINKIEWICZ,2015). τDS,LP = VDS,LP HDS,LPm˙nomDS,LP,inϑDS,LP , τDS,LP dPout DS,LP(t) dt =kDS,LP(m˙ out DS,LP(t) −m˙ in DS,LP(t)) (2.28)
To calculate the output mass flow rate and the temperature of the dead space,
˙ mout
DS,LP(t) and TDSout,LP(t), static relationships are used, seen in Equations (2.29) and (2.30), respectively. ˙ min DS,LP(t) =m˙ out R (t), ˙ mout DS,LP(t) =m˙ in DS,LP(t), (2.29) Tout DS,LP(t) =ADS(P (t)outDS,LP) BDS(P (t)outDS,LP)CDS, (2.30) wherem˙ in
DS,LP(t) is the mass flow rate on the inlet of the dead space. The parameters can be seen at Table A.10on AppendixA.
Chapter 2. System Modelling 24 2.1.5.7 Low Pressure Stages
Much like theHP section, the LPsection’s model is simplified to a static model, as its dynamics is considerably fast comparing to the other components of the steam turbine. TheLPsection is divided in six stages, with three vents after stages two, four and five. Each stage is represented by i on the following equations.
For the calculation of the output pressure for each stage,Pout
LP,i(t), Equation (2.31) is used. PinLP,i(t) =P out DSLP(t), i = 1, PinLP,i(t) =P out LP,i - 1(t), PoutLP,i(t) =P nom,out LP,i Pin LP,i(t) PnomLP,i,in
(2.31)
For the calculation of the output temperature for each stage, Tout
LP,i(t), Equation
(2.32) is used.
TLP,iout(t) =AT(PoutLP,i(t))
BT(PoutLP,i(t))CT (2.32)
For the calculation of the output mass flow rate for each stage,m˙out
LP,i(t), Equation (2.33) is used. ˙ mout LP,i(t) =m˙ nom,out LP,i (t) \sqrt{}
TnomLP,i,in Tin
LP,i(t) Pin
LP,i(t)2 −PoutLP,i(t)2
(PnomLP,i,in)2 − (P nom,out LP,i )2
(2.33) For each stage with a vent, the calculation of the output mass flow rate for each vent,m˙out
LP v,i(t), Equation (2.34) is used. ˙ moutLP v,i(t) =m˙ nom,out LP v,i ˙ mout LP,i=ivent(t) ˙
mnomLP,i=i,outvent(t), (2.34) wherePin
LP,i(t),PoutLP,i - 1(t) are the pressure on the input of each stage and on the output
of the stage before, respectively, Tin
LP,i(t) is the input temperature on each stage and ˙
mout
LP,i=ivent(t) is the output mass flow rate of the stage equivalent to the vents’ stage. The
parameters can be seen at TableA.11 on AppendixA.
2.1.5.8 Mechanical Power
To calculate the mechanical power generated by the steam turbine, the values of the flow rate and temperature for each stage group on the HP andLP sections are
Chapter 2. System Modelling 25 used. Firstly, the enthalpy drop, ∆hout, for each stage group is calculated using Equation (2.35).
∆hout(t) = ∆hnom,out( Tin(t)
Tnom,in 1 − (PPoutin(t)(t)) H−1 H 1 − (Pnom,out Pnom,in) H−1 H ) (2.35)
Then, the theoretical power is calculated on each group of stages,NC(t),
accord-ing to the calculated enthalpy drop, as shown on Equations (2.36).
Ns,k(t) =Nnoms,k knom s,k ηnom s,k ˙ mout s,k(t) ˙ mnoms,k ,out ∆hout s,k(t) ∆hnoms,k ,out , NC(t) = nHP+nLP \sum s\in \{ HP,LP \} ,k\in \{ 1,...,ns\} Ns,k(t), (2.36)
where Tin(t) is the temperature on the input of each stage on theHPor LPsections and Pout(t) andPin(t) are the pressure on the output and the input of each stage on theHP orLPsections, respectively. m˙out
s,k(t) is the output mass flow rate of each stage.
The effective power of the turbine,NT(t), is calculated based on the theoretical
power calculated, as can bee seen in Equations (2.37).
ηe(t) =ηnome ( NC(t) Nnom C )Aη( N nomC NC (t)) 2 , NT(t) =NC(t) ηe(t) ηnom e (2.37)
The parameters can be seen at TablesA.12andA.13on AppendixA. The turbine’s output generated power is the input to the electricity generator and its output steam temperature, pressure and flow rate are the inputs to the condenser.
2.1.6 Electrical Power Generator
Electricity generators convert mechanical power contained on the rotational axis to electrical energy through a magnetic field, based on electro magnetic induction phenomena. The rotor’s spin induces a voltage on the winding terminals, which are connected to loads where electric current is circulated. Although it can be modelled with many states, depending on the rotation of the motor and many variables, like in (SOKÓLSKI; RUTKOWSKI; DUZINKIEWICZ, 2017) and (SOKÓLSKI; RUTKOWSKI; DUZINKIEWICZ,2015), the model used in this work is simplified, since the non-linear dynamics will be modelled directly on the steam turbine.
Chapter 2. System Modelling 26 The modelling of the electricity generator is simplified to a first order linearised model (ZIMMERMANN,2012), thus it is not a phenomenological model like most of the models presented in this chapter, but an empirical model, which is further explained in chapter3. The model’s output is the variation of rotational velocity (MENARIN,2013), ∆ω(t), and it is calculated using Equation (2.38).
τG
d∆ω(t)
dt + ∆ω(t) = NT(t) −ND(t), (2.38) whereNT(t) is the power generated by the steam turbine and ND(t) is the power
de-mand. τG is the model’s time constant, which depends on the type and size of electricity
generator used on the process. The system’s main goal is to maintain the angular veloc-ity at its operating point, which is the frequency of the electric grid. In this case, the frequency of the electric grid is 60Hz and the aim is to maintain ∆ω(t) = 0, in other words, maintain the variation of the grid’s frequency at zero.
2.1.7 Condenser
The condenser is a part of the thermal equipment that allows two fluids with different temperatures to transfer heat between each other in adjacent chambers (FILHO,
2013) so that the hot fluid will change its phase. In this project, it will be used to transfer heat between the steam that exits the turbine and the water at room temperature, so the steam will condensate, turn into water and go through the steam generator again.
In literature several complex models can be found. In (MILIáN et al., 2013), a dynamic model of a condenser was presented as a set of seven non-linear ODE. In (SANAYE; DEHGHANDOKHT,2011), a parallel flow condenser was modelled and much like in (MILIáN et al.,2013), there are many unnecessary states being calculated.
The non-linear dynamics of the condenser, thanks to it being a two-phase system like the steam generator, makes its modelling a complex matter. The model chosen for this thesis is highly simplified as a static model, since the dynamics of the steam generator and turbine are more relevant for the simulation and control calculation than the dynamics of the condenser. Some assumptions were made to calculate the model: there is no pressure drop in the steam side, no sub cooling occurs and the enthalpy of the steam at the condenser output equals the saturated liquid enthalpy at the condensing pressure (PATNODE,2006).
Chapter 2. System Modelling 27 wherem˙w(t) is the cooling water’s flow rate.
U A(t) = UAref ˙ mw(t) ˙ mw,ref 0.8 , N T U (t) = U A(t) ˙ mw(t)Cw , ε(t) = 1 − e - N T U (t), (2.39)
where UA(t) is the heat transfer conductance-area product and NT U(t) is the number of transfer units. The water’s temperature, Tw(t), is calculated using Equation (2.40), wherehs,out is the output enthalpy of the steam.
Tw(t) =
hs,out(t) Cw
(2.40) The heat transfer between the water and the steam and its maximum heat trans-fer possible, Q(t) and Qmax(t) respectively, are calculated using Equation (2.41), where
˙
ms(t) is the steam’s input flow rate, hs,in(t) is the input enthalpy of the steam ( HOLM-GREN,2007) andTw,in(t) is the cooling water’s input temperature.
Q(t) =m˙s(t)(hs,in(t) −hs,out(t)),
Qmax(t) =m˙w(t)Cw(Tw(t) −Tw,in(t))
(2.41)
The efficiency of the condenser is calculated using Equation (2.42).
ε(t) = Q(t) Qmax(t)
(2.42) Equalling the above Equations (2.42) and (2.39), and substituting Equations (2.41) and (2.40), the steam’s output enthalpy value is found and, with its value and the condense pressure of the steam turbine, the output temperature of the steam is calculated (HOLMGREN,2007).
The parameters used in this model can be seen on Table A.14 on Appendix A
and they were chosen based on the process dimensions. While simplified, this model is a good representation of the phase change from steam to liquid water. Its output temperature is one of the steam generator’s input.
2.2 THE FINAL PROCESS
The chosen systems used to model the process in this work are selected based on the complexity of each subsystem, as they have to be complex enough to describe well
Chapter 2. System Modelling 29 Table 2.1: Variables of each equipment
Equipment Manipulatedvariables Disturbances Process variables Solar
collectors field
Oil’s flow rate (˙qscf(t))
Irradiation (I(t)) Ambient temperature (Ta(t))
Oil’s temperature (Tscf(t)) Hot
tank Oil’s velocity(v(t))
Oil’s temperature (Tscf(t)) Ambient temperature (Ta(t))
Oil’s temperature (Tht(t))
Gas
heater Power (Ngh(t))
Oil’s flow rate (˙qht(t)) Oil’s temperature (Tht(t)) Ambient temperature (Ta(t))
Oil’s temperature (Tgh(t)) Cold
tank Oil’s velocity (
v(t)) Oil’s temperature (Tht(t)) Oil’s temperature (Tct(t)) Steam generator Water’s flow rate (m˙c(t)) Heat (Q(t)) Water’s temperature (Tc(t)) Steam’s flow rate (m˙s(t))
Water’s drum level (l(t)) Steam’s quality (αsg(t)) Total water volume (Vsg,w(t))
Steam’s pressure (Psg(t)) Steam’s volume under the liquid level (Vsg,s(t)) Steam turbine Steam’s pressure (Psg(t)) Valve opening (α(t)) Steam’s pressure (Pst(t)) Steam’s flow rate (m˙st(t)) Steam’s temperature (Tst(t)) Steam turbine’s power (Nst(t)) Electricity
generator Demand power (ND(t)) Angular velocity variation
(∆ω(t))
Condenser
Steam’s pressure (Pst(t)) Steam’s flow rate (m˙st(t)) Steam’s temperature (Tst(t))
Water’s flow rate (m˙w(t)) Water’s temperature (Tw(t))
Water’s temperature (Tc(t))
2.3 FINAL REMARKS
This chapter presented the modelling of each system which composes the process, by utilizing mostly energy, mass or momentum balances. The proposed models were developed to solve a trade-off between complexity and process dynamic representation. As will be seen in Chapter5, the simulations results show that the model is accurate enough for the objective of this work. The control theory will be studied on Chapter3
3 MODEL PREDICTIVE CONTROL
Model Predictive Control(MPC) is a set of control strategies which use the process model to predict its future behaviour and thus calculate the control signal by minimizing an objective function (CAMACHO; BORDONS, 2007). It was first developed in the industry, and then its theory was studied in the academia (PATWARDHAN,2014). MPC
algorithms are categories of advanced control substantially used in the industry due to its good response when used in many types of processes, such as processes with delay, with a non-minimal phase, unstable or multi variable (CAMACHO; ALBA,(2013)).
MPCis commonly used in the solar energy generation industry, as it shows good practical results, particularly because it takes into consideration the system’s restrictions (Torrico et al., 2010). Energy generation systems are complex processes, with several subsystems andMPCis capable of controlling the plant globally, while simpler controllers can only operate locally. Usually, Non-linear Model Predictive Control(NMPC) are used since they utilize non-linear models as their prediction model.
There are many studies ofMPCon the solar energy generation field. In (ROCA et al.,2016), a two-layerNMPC strategy is used to produce distilled water using a solar field as heat source. In (GÁLVEZ-CARRILLO; KEYSER; IONESCU,2009) aNMPCis used, as well as a Dead Time Compensator, to control a solar collector field. In (ANDRADE et al., (2013)) a PNMPC is used to control a non-linear model of a solar field, adding a Lyapunov function to the cost function. In (ELIAS,2018) a hybrid model is used to control the focus of solar fields mirrors and in (GIL et al.,2019) a hybrid model is used to determine the optimum operation of a solar membrane distillation facility. In this thesis, aNMPCis used to optimize the energy generated by the solar process shown in Chapter2.
Regardless of the variety of MPC used, they all adopt the same strategy ( BOR-RELLI; BEMPORAD; MORARI, 2017), shown in Figure 3.1. In it, the future outputs in the prediction horizon, N, are calculated using the process model. They depend on the inputs and outputs past values and on the future control signals which will be applied to the process. These future control signals are calculated usually aiming to lead the system’s controlled output to a reference and reduce the control effort. MPC uses a receding horizon strategy, which applies only the first control action from the calculated array, shifts the control horizon to the future and repeats the calculation of the control array in the next sampling time. The first control action is sent to the process and, in the next sample period, this calculation will be repeated and the sequence of control actions will be updated.
